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Forum:Save the oodles?
The idea of oodles is that every collection of oodles is also an oodle, e.g. oodle of all oodles, which in this case contains itself. Unfortunately, there is nothing stopping us from taking exactly the oodles which do not contain themselves, and forming an oodle out of them. The question if this oodle contains itself leads to Russell's paradox. The theory of oodles is equivalent to naive set theory, which is a well-known example of inconsistent theory. The way of saving that theory is making restrictions to allowed sets. However, this is a syntactic restriction which can be considered "unnatural". Do any of you have ideas on how to modify oodles so that they become consistent? LittlePeng9 (talk) 04:58, October 20, 2014 (UTC) :axiom of foundation? it's vel 09:24, October 20, 2014 (UTC) ::It won't help. Let me explain the issue in detail: let CO denote the "collection of oodles" axiom/property: any collection of oodles is an oodle. As I explained, CO is inconsistent all by itself, because it implies that Russell's paradoxal oodle is a valid structure. Adding AF can only possibly make it worse - we still can derive contradiction from CO. It's well known fact that extending inconsistent theory can't magically make it contradiction-free. What we need is a weakening of CO or some other way around the problem. LittlePeng9 (talk) 10:00, October 20, 2014 (UTC) :::so let's restructure oodles so they can never contain themselves it's vel 15:36, October 20, 2014 (UTC) ::::Although not so easily, we can derive Russell-like paradox from this too: let S be oodle of all oodles A which do not contain oodle B which contains A. Now let S' be set of all singletons of elements of S. Question: Does S' contain {S'}? Of it does, then it doesn't, but if it doesn't, then it does. We would have to refine this idea a lot. LittlePeng9 (talk) 15:40, October 20, 2014 (UTC) :::::oops, i said that wrong. *let's restructure oodles so they respect AF. it's vel 15:43, October 20, 2014 (UTC) (removing indentation) So, as I said on the chat, if we had oodles satisfy the (analogue of) axiom of foundation, we would have naturally induced notion of rank of the oodle: * Empty set is its own rank. * If some element of the oodle has rank \(\alpha\) such that \(\alpha\) either is rank or contains ranks of all other elements of the oodle, then it has rank \(\alpha\cup\{\alpha\}\). * If no such \(\alpha\) exists, then rank of oodle is union of ranks of its elements. Note that the definition avoids word "ordinal". Now it's simple to show that every oodle has rank - otherwise, let \(A_1\) be oodle without rank. If all its elements had rank, it would have a rank, so there must be oodle \(A_2\in A_1\) without rank. Continuing, we can have infinite sequence \(A_1\ni A_2\ni...\), existence of which contradicts AF. It's also quite simple to see that oodles of rank \(\alpha\) or element of it are exactly the collections of oodles with rank being element of \(\alpha\). This way we can have natural continuation of Von Neumann's hierarchy to \(V_\alpha\) of oodles with rank \(\alpha\) or element of it. LittlePeng9 (talk) 16:54, October 20, 2014 (UTC) :hell yes! it's vel 18:55, October 20, 2014 (UTC)